A NEW IMPROVEMENT EUCLIDEAN ALGORITHM

Neural, Parallel, and Scientific Computations, 26, No. 3 (2018), 355-362

ISSN: 1061-5369

A NEW IMPROVEMENT EUCLIDEAN ALGORITHM FOR GREATEST COMMON DIVISOR. I ANTON ILIEV1 , NIKOLAY KYURKCHIEV2 , AND ASEN RAHNEV3 1,2,3

Faculty of Mathematics and Informatics University of Plovdiv Paisii Hilendarski 24, Tzar Asen Str., 4000 Plovdiv, BULGARIA ABSTRACT: In this note we gave new realization of Euclidean algorithm for calculation of greatest common divisor (GCD). Our results are extension of results given in [1]–[26], [41]–[64]. For computer implementation Visual C# 2017 programming environment is used. We optimize about 40% and about 10% Knuth’s recursive and iterative algorithms respectively. AMS Subject Classification: 11A05, 68W01 Key Words: greatest common divisor, Euclid’s algorithm, Knuth’s algorithm, improvement algorithm, reduced number of iterations Received: July 1, 2018 ; Accepted: Published: November 27, 2018. doi: Dynamic Publishers, Inc., Acad. Publishers, Ltd.

November 23, 2018 ; 10.12732/npsc.v26i3.10 https://acadsol.eu/npsc

1. INTRODUCTION Our work is next part of research in [27]–[40]. In most of these papers and book [33] we already received some improvements of Euclidean algorithm, extended Euclidean algorithm, multiplicative inverse algorithm, extended Euclidean algorithm using SGN function, algorithm for continued fractions.

2. MAIN RESULTS Now we set the task to optimize Euclidean GCD algorithm. For testing we will use the following computer: processor - Intel(R) Core(TM) i7-6700HQ CPU 2.60GHz, 2592 Mhz, 4 Core(s), 8 Logical Processor(s), RAM 16 GB, Microsoft Windows 10 Enterprise x64.

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Let a>0 and b>0 be natural numbers. We propose new organizing of the Euclidean algorithm: Algorithm 1. if (a > b) while (((a %= b) != 0) && ((b %= a) != 0)); else while (((b %= a) != 0) && ((a %= b) != 0)); gcd = a + b; Its recursive implementation is: Algorithm 2. static long Euclid(long a, long b) { long r, u = b; if ((r = a % b) != 0 && (u = b % r) != 0) return Euclid(r, u); else return r + u; } We will compare Algorithm 1 with Knuth’s iterative Algorithm 3. while (b > 0) { ob = b; b = a % b; a = ob; } gcd = a; and Algorithm 2 with Knuth’s recursive Algorithm 4. static long Euclid(long a, long b) { if (b == 0) return a; long r = a % b; return Euclid(b, r); }

3. NUMERICAL EXPERIMENT Part 1. long a, b, gcd, ob, d = 0; for (int i = 1; i < 100000001; i++) { b = i; a = 200000002 - i; //here is the source code of every one of algorithms 1, 3 //and calling of recursive algorithms 2 and 4 d += gcd; } Console.WriteLine(d); Part 2. We will use the task from Part 1. where we swapped the values of ‘a’ and ‘b’. Part 3. Average time of performance

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Figure 1: Knuth’s recursive (1 – red color), Iliev–Kyurkchiev–Rahnev recursive (2 – green line)

EN = (Part 1.AlgorithmN + Part 2.AlgorithmN ) / 2, where N = 1 to 4 denotes using of Algorithms 1 to 4.

Both recursive implementations can be called by: if (a > b) gcd = Euclid(a, b); else gcd = Euclid(b, a); The reader can see the advantages of new improvement suggested by us (see Fig. 1 and Fig. 2 for recursive and iterative implementations respectively). We will note that this improvement (see Algorithms 1 and 2) gives a better performance even than previous results in [27] and [28] concretely for Euclidean algorithm.

ACKNOWLEDGMENTS This work has been supported by the project BG05M2OP001-1.001-0003-01 “CoE on Informatics and ICT” supported by the Operational Programme “Science and Education for Smart Growth”.

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Figure 2: Knuth’s iterative (1 – red color), Iliev–Kyurkchiev–Rahnev iterative (2 – green line) REFERENCES [1] A. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Computer Algorithms, 1st ed., Addison-Wesley, Boston (1974). [2] A. Aho, J. Ullman, J. Hopcroft, Data Structures and Algorithms, 1st ed., AddisonWesley, Boston (1987). [3] A. Akritas, A new method for computing polynomial greatest common divisors and polynomial remainder sequences, Numerische Mathematik, 52 (1988), 119– 127. [4] A. Akritas, G. Malaschonok, P. Vigklas, On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials, Serdica Journal of Computing, 9 (2015), 123–138. [5] M. Alsuwaiyel, Algorithms: Design Techniques and Analysis, Lecture Notes Series on Computing, revised ed., World Scientific Publishing Company, Hackensack (2016). [6] L. Ammeraal, Algorithms and Data Structures in C++, John Wiley & Sons Inc., New York (1996). [7] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York (1976).

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